Since the operator \(D\)  is invariant under the diagonal action of \(K\)  given by adjoint actions on \(U(\mathfrak {g})\)  and \(C(\mathfrak {p})\) , it follows that \(\mathop {\hbox{Ker}}\nolimits D\) , \({\rm Im}\, D\)  and thus \(H_D(\pi )\)  are all \(\widetilde{K}\)  modules.
Vogan conjectured that whenever non-zero, the Dirac cohomology \(H_D(\pi )\)  should reveal the infinitesimal character of \(\pi \) . This conjecture was proven by Huang and Pandžić in 2002 [1]}. Let us state its tiny extension to the current setting (recall that being in the Harish-Chandra class, the group \(G\)  may be disconnected). We regard \(\mathfrak {t}_f^*\)  as a subspace of \(\mathfrak {h}_f^*\)  by extending the linear functionals on \(\mathfrak {t}_f\)  to be zero on \(\mathfrak {a}_f\) .
